Small-gain theorem

In nonlinear systems, the formalism of input-output stability is an important tool in studying the stability of interconnected systems since the gain of a system directly relates to how the norm of a signal increases or decreases as it passes through the system. The small-gain theorem gives a sufficient condition for finite-gain ${\displaystyle {\mathcal {L}}}$ stability of the feedback connection. The small gain theorem was proved by George Zames in 1966. It can be seen as a generalization of the Nyquist criterion to non-linear time-varying MIMO systems (systems with multiple inputs and multiple outputs).

Feedback connection between systems S₁ and S₂.

Theorem. Assume two stable systems ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ are connected in a feedback loop, then the closed loop system is input-output stable if ${\displaystyle \|S_{1}\|\cdot \|S_{2}\|<1}$ and both ${\displaystyle S_{1}}$ and ${\displaystyle S_{2}}$ are stable by themselves. (The norm can be the infinity norm, that is the size of the largest singular value of the transfer function over all frequencies. Also any induced Norm will lead to the same results).[1][2]